Until a few months ago, Ms. Yellen had been widely seen as President Barack Obama’s no-brainer choice to lead the central bank.

She would be the first woman to head the Fed since it was created a century ago, relieving criticism about Mr. Obama’s track record in nominating women to top posts. Her Democratic credentials are as good as they get for a policymaker with her experience, including years as a top White House economic official in the late-1990s boom years. Her selection would mark continuity at the Fed, maintaining the Ben Bernanke policies that are supporting a weak U.S. economy amid destructive budget battles between Congress and the White House.